Finding Domain and Range

Domain and range: what goes in, what comes out

• Domain: all $x$-values for which $f(x)$ is defined (what can you plug in?)
• Range: all $y$-values that $f(x)$ actually produces (what comes out?)

Two common restrictions

1. Can't take the square root of a negative: set the expression under $\sqrt{\phantom{x}} \geq 0$.
2. Can't divide by zero: set the denominator $\neq 0$.

Step-by-step: Find domain and range of $f(x) = \sqrt{4 - x^2}$

Domain

Step 1 — Set the radicand $\geq 0$:

$4 - x^2 \geq 0$

Step 2 — Solve the inequality:

$x^2 \leq 4$ $-2 \leq x \leq 2$

Domain: $[-2, 2]$.

Range

Step 3 — Find the minimum and maximum output values.

• At $x = 0$: $f(0) = \sqrt{4} = 2$ (maximum)
• At $x = \pm 2$: $f(\pm 2) = \sqrt{0} = 0$ (minimum)
• $f(x)$ is always $\geq 0$ (square root is non-negative)

Range: $[0, 2]$.

Geometric insight

$y = \sqrt{4 - x^2}$ means $y^2 = 4 - x^2$, or $x^2 + y^2 = 4$. This is a circle of radius 2 centered at the origin. Since $y = \sqrt{\ldots} \geq 0$, we get only the upper semicircle.

Quick reference: common domain restrictions

• $\sqrt{f(x)}$: need $f(x) \geq 0$
• $1/f(x)$: need $f(x) \neq 0$
• $\ln(f(x))$: need $f(x) > 0$

Try it in the visualization

Select from several functions. The domain is highlighted on the x-axis (blue shading), and the range is highlighted on the y-axis. The graph only appears where the function is defined — endpoints are marked.